Definition:Either-Or Topology

Definition

Let $S = \closedint {-1} 1$ be the closed interval on the real number line from $-1$ to $1$.

Let $\tau = \set {U \in \powerset S: \paren {\set 0 \nsubseteq U} \lor \paren {\openint {-1} 1 \subseteq U} }$ where:

$\powerset S$ is the power set of $S$

$\lor$ is the inclusive-or logical connective.

Then $\tau$ is the either-or topology, and $T = \struct {S, \tau}$ is the either-or space

That is, a set is open in $\tau$ if and only if it does not contain $\set 0$ or it does contain $\openint {-1} 1$.

Also see

• Either-Or Topology is Topology

• Results about the either-or topology can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $17$. Either-Or Topology